Outline of Thursday , Apr 2 class

Next class we will begin our investigations of 1-D QM. Specifically we will briefly study momentum eigenstates for free electrons, and then focus on the ground states of 4 different 1-D potentials and begin calculating expectation values of things like p and x for those (and p^2 and x^2).

The 4 potentials will be: infinite square well, harmonic oscillator, attractive delta-function and finite square well. (Note that two of them are actually limiting cases of the 4th one.) We will use the Sch.-wave eqn to solve for the g.s. in each case with appropriate boundary conditions. Interestingly, I think the boundary conditions appear quite differently in each case. You may want to look at the delta-function potential, and how the ground state emerges for that before class. Then, as mentioned above we will start calculating stuff; hopefully that will begin to meet your need for a hard-core quantum class that goes well beyond 101. This will form the basis for most of the second HW assignment, which will be posted later (and which will be due Thursday April 9). (Also later I will post a more nuanced outline for the first part of the class.)

Regarding the book (Liboff), and the implied question, "what should I read from the book for this?", here are my thoughts. This book is a new one being used for the first time this year. It seems like a very nuanced and thoughtful book. It is considerably longer and pitched at a higher level than Griffiths, the book that was used in this class for past decade or so. One of its strengths, as you will see, is that it is an excellent reference. The notation seems appealing, and some signature problems, like the spreading of a gaussian wave-packet, are worked out nicely. It some other regards, such as pedagogy, brevity and organization, you may find that "traditional books", like Griffiths, seem more aware of the readers experience and are more accessible. This is to some degree a matter of personal preference. I would recommend: coming to class, following this blog, and taking the initiative to find what book works best for you. There are interesting reviews and comparisons of quantum books at amazon that provide some perspective that may be useful.

http://www.amazon.com/Introductory-Quantum-Mechanics-Richard-Liboff/product-reviews/0805387145/ref=cm_cr_dp_all_helpful?ie=UTF8&coliid=&showViewpoints=1&colid=&sortBy=bySubmissionDateDescending

http://www.amazon.com/Introduction-Quantum-Mechanics-David-Griffiths/dp/0131244051

Outline and expectations ...

Welcome to physics 139a. This class weaves together 3 themes which could be loosely characterized as:

1) quantum formalism, postulates, mathematical underpinnings and structure,

2) quantum physics in 1-dimension (mostly bound states),

3) quantum physics in more than 1-d (2 & 3), focusing particularly on the hydrogen atom with its interesting and unusual degeneracies, etc.

One could imagine covering 1) before 2) or 3); but that would probably be overly formal, slow, and pedagogically disastrous. Instead we will try to weave the formalism into 2) and 3) so that we will have concrete examples to work with along the way, and so that we will be able to get as far as we need to in this 1 quarter, 10 week class. The material on the hydrogen atom at the end of the quarter is important both for people planning to take 139b and for those taking this as their last class in quantum.

Quantum physics is amazing and provides the underpinning for most of physics, chemistry and more. It is primarily the story of the electron. We endeavor to develop an integrated mathematical and intuitive understanding of this magnificent small creature which, when confined, acquires significant kinetic energy. This simple and unusual trait plays a central role in many quantum phenomena of importance.
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Background:

Linear algebra is exceedingly important to QM (see problems in HW post below). The more you know about finite-dimensional linear algebra (basis sets, inner products, spanning, linear independence,...) the better. As the book (Liboff) discusses in 4.4, QM is based on Hilbert spaces, which are a infinite dimensional analogue to finite-dimensional vector spaces. Most of the same concepts carry over. Speaking of finite dimensional vector spaces, we will use those too, and an intimate familiarity with hermitian matrices and their eigenvalues and eigenvectors is essential.

Where we will start:

We will start with the wave equation (now called the Schrodinger eqn.) for electrons. With that and other starting postulates we will do separation of variable, talk about time dependence, normalization, expectation values, ... and get into some examples with the infinite square well and other 1-d potentials including: the attractive delta-function, the finite sq well and the harmonic oscillator. In addition to learning how to calculate things, it is valuable for you to develop a sense of aesthetics. What is appealing or useful with each of these potentials?; what are their flaws, weaknesses or ...?

For more, see the April 1 post, Outline and road map.

Homework 1 with solutions: please comment...

Note that for problem 5 the most important part is the last step where we use the eigenvectors to match initial conditions and thereby infer time dependence (see last paragraph of #5).

These review-related problems (0-5) are due in class Thursday, April 2. You can have more time for problem 6 if you like. Comments and questions are welcome and encouraged.

PS. (added 3-29) If you put your picture in your profile, that will help me get to know people. (You can always change it later, after a few weeks?, if you want.)

#0. What is your sense of the basic postulates or starting point of quantum physics? (no need to do a big research project. Just what you know now is fine.) What is quantum physics? What does it tell us? What is it important to?

#1. (Wave eqn problem) a) Write the wave equation for a string under tension. (Wave eqn refers to the thing with the spatial and temporal derivatives and the wave-speed.) b) To explore a little, express the wave-speed in terms of T (tension) and m/L (mass density) and c) rewrite the wave equation with just m or m/L on the "right-hand side" (with the time derivatives) (and T moved over to the other side).

d) Compare this to the 1-dimensional, time-dependent Schrodinger wave equation (with no potential), if you are familiar with that.

#2. (String problem) Consider a string of mass m and stretched between two posts a distance L apart.

a) What are the solutions of the wave equation for the transverse motion of this string? (I guess what I am really asking here is what are the normal modes.)

b) For an initial string displacement y(x) (at t=0), write an expression for the time dependent displacement, y(x,t).

c) Create and simple, but non-trivial, example of time-dependent displacement.
Discuss and elaborate.

3. (Linear algebra) Write one or more well-constructed paragraphs expressing your understanding of linear algebra. Emphasis on concepts and words such as "basis", linear independence, spanning, orthogonality, inner-product and normalization is encouraged.

4. (Hermitian) Discuss Hermitian matrices. What are they? What are they characteristics of their eigenvalues and eigenvectors? What is an eigenvector? What is an eigenvalue?
Please discuss and explain, and provide functional definitions and give an example or two.

5. Consider a linear system (1 dimensional) of 2 equal masses, m, and three springs.

||-----------m--------m-----------||

For our purposes, let us assume that the two outer springs are identical (spring constant k) but that the inner spring, the one connecting the two masses together, can be stronger, weaker or the same as k.

a) (What are our purposes? Why is it helpful for the masses and the outer springs to be the same? (The answer can be one word, though more nuance is welcome.)

Assume that there is an equilibrium configuration for the system such that all 3 springs are at their natural length and let us call the deviations from that: x1(t), for the position of the left hand mass, and x2(t) for the right-hand mass. Suppose that at t=0 the left-hand mass at x1(o)=1 meter (or x_0 if you prefer) and the other mass is at either zero or also 1 meter. (These are two different problems with 2 different solutions.)

b) Solve for x1(t) and x2(t).

(hint: How do you approach/begin this problem? Please comment below.)

6. What is your understanding of the origin of the size of the hydrogen atom. (Just H. No discussion of multi-electron atoms please.) (Size, not mass.)

Discussion section scheduling for Physics 139a

Hi. Welcome to physics 139a. Soon I will post a course outline and the first homework here. For now, we would like to ask for your input regarding the discussion section for this class.

To provide some context, the discussion section is not generally mandatory (you probably know that), but it is a valuable resource for help with HW, concepts, etc. There will be some difficult HW in this class and quantum can be challenging and/or confusing the first time you go through it in a mathematically serious way (though once you understand it there is a simplicity that is appealing). Anyway, I think you may want to avail yourself of the help the discussion section provides. Additionally, I have been thinking about the practicality of doing some testing (midterm or quizzes) in a few of the discussion sections this quarter. (There would be prior notice...)

Currently, the discussion section is scheduled for 7:00 to 8:30 PM Wednesday in ISB 231. (That is a good room, I think, because the tables allow for interaction.) Another possibility is Tuesday at 4:00-5:30 PM, in ISB 235 (also a good room; just a little smaller).

Please let us know you opinion regarding that, and what day and time would be best for you. Please post something, as a comment to this blog, to let us know that you have seen this. If you have no opinion, or are not sure yet, that's fine, but please post something.

In case you want to get a head start, look for more posts over the next few days and this weekend. See you Tuesday!
-Zack